International Trade, Entrepreneurship, and Monopolistic Competition: German Hidden Champions and Global Markets

The Chicago School of Monetarism dominates the debate on the competition models. In the last decades, the pendulum has swung back towards the German historical school of economics by List, Schumpeter, Gutenberg, and Simon. The new trade theory by Paul Krugman is the standard in analysis of economies of scale and product differentiation in intra-industry trade. Chamberlin’s product differentiation concept and monopolistic competition theory are included in Krugman’s theory. Diversity matters in B2B-business in global markets. About 100,000 multinational corporations (MNCs) dominate the international trade of commodities worldwide but they need partners as suppliers of complementary products, services, and technologies. German hidden champions (HCs) that consist of over 1,300 mid-sized firms (Mittelstand) have revolutionizes the managerial economics. In terms of Peter Drucker, striving for leadership is the greatest gamble. Their success rate has been about 90% as Hermann Simon has reported. Their success recipe is the humble choice of markets segments, mainly B2B-industries, to make good business in any kind of goods and articles, not to follow trends or hit lists. The dilemma in EU-27-countries is that only Germany has succeeded to develop its own management doctrine initiated by Friedrich List and modified by Joseph Schumpeter. List argued that economic policy had to be adapted to the needs of specific nations to create the national system of innovation. Germany is the most diversified country worldwide. German small and medium-sized firms (Mittelstand) are globally oriented. Germany’s family business is the unbelievable success story. Alfred Chandler has been a highly influential business writer. Because of Chandler’s view, the personal capitalism was generally thought to be the old-fashioned model in comparison to the stock market capitalism. The Germany case is the strong evidence on the fact that Chandler’s famous conclusion may be wrong. The family-ownership is perhaps the most sustainable governance model in the global economy. The US is the winner of Chandler’s stock market capitalism, but the majority of US firms are domestic-market-oriented. In Germany about 100,000 mid-sized firms have experiences about FDI-operations.

wealth is its productive power, rather than its current exchange values; and (2) private economic interests must be subordinated to the strengthening of the nation. List advised Germany and the U.S. to develop education, railways, and technology. List's ideas have been the basis for three economic miracles (Wirtschaftswunder) in Germany: (1) Zollverein from 1834 to 1919, (2) West Germany from the 1950s to the 1960s, and (3) globalization of Germany since 1990s, related to the completion of the vision of European economic integration by Bundeskanzler Konrad Adenauer (Wintle, 2002).
Schumpeter did not accept the interpretation of the Walrasian process of adaptation (Samuelson, 1947) in which economic agents (suppliers, consumers, etc.) are rational optimizers. Schumpeter (1934Schumpeter ( , 1939 analyzed economic development and business cycles. He claimed that an innovative transformation in a capitalistic society is a slow and conflict-ridden process (Schumpeter, 1942(Schumpeter, , 1994Andersen, 2009) in which the entrepreneur is the hero of the drama (see Lintunen, 2000). Schumpeter introduced the concept of temporary monopoly profit as the lifeblood of innovativeness. As Frank Knight, the founder of the Chicago School (Institutional), Schumpeter was convinced that the so-called normal profit by neoclassical economics is not a satisfactory compensation of innovations and risk-taking under market circumstances of immeasurable unknown or true uncertainty (Knight, 1920). Innovations are the main driver of economic progress and welfare. There are two different innovation mechanisms (Lahti, 2010(Lahti, , 2012: (1) Creative destruction is associated with radical innovations entering unexplored market where there are low entry barriers for new entrants that utilize the public domain knowledge. This is a microeconomic process by its nature and has strong macroeconomic implications for economic growth (Aghion & Howitt, 1998;Romer, 1989Romer, , 1990. (2) Creative accumulation is associated with organized R&D that MNCs carry out innovation along established technological trajectories. MNCs dominate about 2/3 of R&D investments and about 80% of international commodity trade worldwide. As market leaders, MNCs have impact on industry life cycles and market structures (Scherer, 1999) and on rules of the game in global markets (Arrow & Hahn, 1971). Schumpeter (1934) argued against Marshall who in the early 20th century was the most intelligent economist of the British Economic School. Marshall (1920) exerted great influence on the development of economic thought and initiated the revolution of marginalism. His contributions include two concepts: marginal utility and representative firm. Marshall made the distinction between (1) external economies-economies of scale that depend on the firm's ability to grow to the optimal scale, and (2) internal economies-economies of scope that depend on the firm's resources, organization, and management efficiency. Schumpeter never denied Marshall's merits as the forefather of managerial economics. His main concern was that Marshall excluded entrepreneurial (innovative) behavior from the frame of neoclassical economics. When Marshall believed in the rational mind of managers, Schumpeter gave more credits to human agency. Chamberlin (1933) and Robinson (1933) are regarded as the parents of the modern study of imperfect competition. Chamberlin (1933Chamberlin ( , 1937Chamberlin ( , 1948Chamberlin ( , 1951Chamberlin ( , 1957Chamberlin ( , 1965 provided the building block for Schumpeterian analysis of market dynamics by his theory of monopolistic competition (Brue & Grant, 2008). Their collaboration contributed a lot to the modern economic thinking. Chamberlin's main contribution was the highly intellectual theory of monopolistic competition that is widely applied as the foundation of two important doctrines of international business: marketing (parameter) theory and the Structure-Conduct-Performance (SCP) paradigm.
According to a narrow definition, industries with market structures similar to monopolistic competition include e.g. clothing, shoes, and services. According to a broad definition, it is possible to include most of the B2B-industries in international intra-industry trade. Monopolistic competition is the most common model of competition in globalizing markets and 60% of the total (Figure 1). In Krugman's writings (1979,1980,1981) on intra-industry trade, monopolistic competition and product differentiation are the main drives of trade. So Chamberlin's contribution has been fundamental to the trade-driven global economy, although not accepted by most of orthodox economists. Germany has nearly four million firms that have taken advantage of monopolistic competition even in B2B industries that are commonly assumed to be stacked in the keen prices competition and in the continuous cost-minimization according to Marshall's idea. The main hypothesis of this article is: German HCs are winners of the B2B-industries, because they have innovated unique business models that in market behavior follow the key principles of monopolistic competition.

Monopolistic Competition: Realistic View of Global Competition
A monopolistically competitive firm generates brand loyalty and other elements of differentiating marketing activities and thus faces a downward sloping demand curve. Due to the wide availability of substitutes, the demand curve in various globalizing markets tends to be relatively elastic. In a monopolistic competitive industry, there are many firms or many competing products that are close substitutes in nature and many specialty brands. The differences in the brands are either actual or perceived. In the short run, the firm will maximize profit in the point where marginal cost (MC) equals marginal revenue (MR). Average revenue may exceed average cost (AC). In the short run, firms may make supernormal profit ( Figure 2).
Super-normal profits attract new entrants, which shifts the demand curve for existing firm to the left. In the long run, supernormal profits are competed away by new entrants. This shifts the average revenue curve to the left until eventually average revenue (AR) is equal to AC and the firm will earn the so-called normal profit. Unlike a perfectly competitive firm that produces at the minimum point of its total AC curve, a monopolistically competitive firm produces at the point where AR equals MC that is the long-run equilibrium ( Figure 3). In monopolistic competitive markets, none of these firms faces the entire demand curve in the way a monopolist would do, but each does have some power to set prices.  In international trade, there are only few products that in broad terms are homogeneous. Therefore, the monopolistic competition theory is useful. Chamberlin's most intelligent idea is that monopolistic competition includes characteristics of two (orthodox) models of competition: perfectly competition and monopoly. Monopoly was exposed by Antoine Cournot (Sandmo, 2011) and elaborated by Marshall (1920). Chamberlin's idea is that any different strategy of a firm's product gives it to an edge of monopoly. Schumpeter's idea is a temporary monopoly profit that is related to an edge of monopoly in innovations. Chamberlin argued also that marketing costs are not a part of the cost of production, but are incurred to increase the sales of the given product and, thus, they affect the demand curve. In the same way, R&D costs shift the demand curve, although the impact is more dynamic. In the model (see Figure 2 and 3), a monopolistically competitive firm faces a downward sloping demand curve in the same way as a monopoly firm (Goodwin, Nelson, Ackerman, & Weisskopf, 2009). Unlike a monopoly market, a monopolistically competitive market is open for entry of firms (Krugman & Wells, 2009) and, thereby, contestable in the long run, comparing to Baumol (1982).
International trade has contributed strongly to the GDP of nations. The export-led growth has been a key part of China's success during past decades. Chamberlin's most visionary argument is: Monopolistic competition is welfare enhancing, not reducing since a society gets as much product diversity as it is willing to pay for (Bellante, 2004). Diversity matters in global markets. As expected, Chamberlin challenged the leading neoclassical economists by claiming that most of the economic situations are composites of both competition and monopoly, and that, wherever this is the case, a false view is given by neglecting either one of the two forces and regarding the situation as made up entirely of the other (Chamberlin, 1937(Chamberlin, , 1965. In a monopolistic competitive industry, there are relatively many products that are close substitutes not perfect substitutes. The differences among competing products are either actual or perceived. In global markets, there are potentially millions firms supplying products that are close substitutes. Therefore, perfect competition is not possible and monopoly is temporary as both Chamberlin and Schumpeter have claimed.  Chandler (1962Chandler ( , 1990) emphasized the role of general management in the transformation of a firm's strategy and structure (Kreikebaum, 2000;Albach, 1980). In Gutenberg's solution, the individual price-sales function is doubly kinked and a firm has its own monopolistic scope of the price-sales-function to plan its marketing parameters, without having to fear reactions of competitors (Figure 4). When a firm leaves the monopolistic scope, the price is dictated by the market (Pindyck & Rubinfeld, 2009). Gutenberg's price theory has been important for German firms that are aware of how import price positioning is in global markets. The monopolistic scope concept is genius and visual even for small and medium-size enterprises (SMEs). The success story of German HCs (Simon, 2009) is probably based on their unique applications of the ideas of monopolistic competition.

Industrial Organization (IO) Economics
The development of industrial organization (IO) economics as a separate field of micro-theory owes much to two Harvard professors: Edward Chamberlin and Edward Mason (1957), and Joe Bain who was doctorated at the Harvard. The IO is a broad field of studies that are built on the theory of the firm by examining the structures and boundaries of industries and markets (De Jong & Shepherd, 2007). The IO adds multiple real-world imperfections to the competitive models, such as transaction costs (Coase, 1960(Coase, , 1987(Coase, , 1988(Coase, , 1998, limited information (Simon, 1960(Simon, , 1979, and barriers to entry of new firms (Bain, 1951(Bain, , 1956) that are associated with imperfect competition models. Frederic Scherer, a Harvard professor, focused on dynamics in economics, e.g. technological change, leading to numerous publications. In 1970, he summarized the state of art of the Structure-Conduct-Performance (SCP) paradigm in his book of the topics (Scherer, 1970) that is widely thought to be the best book of the SCP ever written. The SCP paradigm assumes that the performance of an industry is determined by how firms in that industry conduct their activities in terms of the market structure. Scherer and Ross (1990, p. 5) summarized the relevant framework for the analyses of SCP ( Figure 5).
The SCP is mainly developed by Bain (1951Bain ( , 1956 to offer the causal explanation for the firm's conduct and performance in imperfect markets. The SCP paradigm revolutionized the IO studies. For the first time, it was possible to make empirical and statistical analyses of how firms are really behaving. The SCP finding was the key to the implementation of the anti-trust legislation in the US from the 1950s to the 1970s. The SCP provided new tools for the US governmental agencies, and, later, to the EU commission, to verify the antitrust rules. The main hypothesis of the SCP is: The degree of seller concentration is inversely related to the degree of competition (Bain, 1956). Barriers to entry, e.g. economies of scale, are the sets of economic forces that create disadvantages to firms that attempt to enter the market. The SCP is the most powerful analytical tool since (1) it is straightforward in its line of reasoning, and (2) it is highly useful in the identification of structural characteristics (Jones & Sufrin, 2010).
In the 1970s, Caves began to modify the Chamberlin-Mason-Bain-Scherer SCP model of structure and performance to include differing positions of strategic groups of firms within industries. He redefined entry barriers (Bain, 1956) to mobility barriers that are persistent structural features, not only at a firm level, but also at a strategic group level, that give rise to structural/strategic and asymmetric mobility barriers protecting a given strategic group from the entry of potential rivals and, thereby, permitting performance differences among groups and, hence, among firms. He guided a serious of dissertations: Hunt (1972), Newman (1973), and Porter (1973). In the global markets, firms try to find out better positioning inside/in-between industries, which leads to structural changes in the whole industry in terms of Schumpeter. Parallel to that, the strategic management doctrine advises firms to make proactive decisions of their products, markets, and technologies (Hoskisson, Hitt, Wan, & Yiu, 1999).
In the Purdue University, Dan Schendel and Arnold Cooper initiated the -brewing‖ studies (Hatten, 1974;Patton, 1976) to explore empirical links among organizational resource choices, interpreted as strategy, and firm's performance (Rumelt, Schendel, & Teece, 1991). The strategic choice approach by Purdue-studies assumes that systematic similarities and differences exist among firms as a result of strategic resource choices, i.e. decisions to invest in assets which are often difficult and costly to imitate (McGee & Thomas, 1986. While the Harvard studies relied on the cross-sectional data in econometric analyses, Purdue-studies used time-series data to draw valid inferences about the performance differences and patterns of competition of a single industry. A new contribution was the use of numerous variables linked to strategy to identify competitive or strategic groups selected within the context of the particular industry under study (Cool & Schendel, 1987). The bottom-up approach by the Purdue-studies is suitable to even small firms, whereas the new Harvard approach is tailored to MNCs. The dilemma of the most studies under the IO is the methodology selected. Using solely cross-sectional data is not a correct approach since strategic groups are highly dynamic. According to the Finnish IO studies by Lahti (1983), Salimä ki (2003), Killström (2005), and Luukkainen (2012), mobility barriers can explain some of sustainable performance differences among strategic groups and, performance differences exist among firms holding identical strategic positions within an industry (Carroll, Pandian, & Thomas, 1994). In most of IO studies, the contribution of a strategic group to a firm's economic performance is weak (Thomas & Venkatraman, 1988). The Finnish IO studies provided evidence that strategic groups are real, and not only constructions of researchers. This depends on the fact that the Finnish researchers have long experience of industries under investigation. The main findings are that the modern SCP/IO is highly useful to many kinds of industry and market analysis, but a careful analysis takes time.
Luukkainen (2012) analyzed the greenhouse sector in Finland and did a massive empirical study. He collected data of 121 greenhouse companies covering 11 years (1998)(1999)(2000)(2001)(2002)(2003)(2004)(2005)(2006)(2007)(2008) and combined many possible sources of data. Together, he did 1,331 firm level analysis. In his study, the strategy process was applied to the greenhouse sector. In contrast to previous studies, Luukkainen's production function was employed in the identification of strategic groups within the field of operation. These groups occupy a central position in the analysis of the industrial context. The production function was used to simulate the system dependency of different combinations of productive input and output factors. The price and the capacity utilisation ratio of productive inputs are crucial as the firms' aims to achieve optimal results in its business environment. Luukkainen noticed that the profitability of the sector declined throughout the 11-year research period due to the import competition and a simultaneous rapid increase in the production costs. The rapid increase in production potential in Finland has resulted in overproduction in the small domestic market, and sufficient resources have not been channeled to the development of export.

New Institutional Economics (NIE)
The New Institutional Economics (NIE), coined by Williamson (1985Williamson ( , 1987, has its roots in the ideas of Ronald Coase. Like Schumpeter and Chamberlin, Coase (1960) questioned the relevance of the orthodox economics, because the language of mathematics due to the demands for determinacy is not designed to deal with complex semantic meanings (Samuels, 1989). Coase argued against the misuse of formalism in neoclassical economics that masks underlying contradictions in the theory. The lack of relevance to solve real world problems is the consequence. Following the idea of agency theory, human behaviour is not rational (Eisenhardt, 1989). Bounded rationality, moral hazards, and risk awareness characterize human behaviour more than assumptions of perfect rationality. Williamson discussed ex-post costs of transacting and Coase discussed the ex-ante costs of transacting.
The transaction cost concept is useful in global contexts. Ex-post contractual failures are salient when transaction costs are significant because of a firm's asset specificity and irrationality in human decision-making. Investments in specific assets need safeguarding, if opportunism arises aimed at the ex-post appropriation of income streams which are generated by the underlying specialized assets, e.g. licensing of patents or franchising brands. In principle, safeguarding against contractual hazards, such as an aggressive patent trolling activity, is difficult to implement in the international context without comprehensive contracting and high legal (transaction) costs. Patent trolling may be highly rewarding at least temporarily. The NIE defines the firm as the governance structure or the nexus of contracts-a view that contrasts fully with the orthodox theory of the firm (Williamson, 1990(Williamson, , 1991. The NIE's notion of contractual function of a firm is useful in global markets where a firm's function is its efficiency in market transactions not only in production as (Coase, 1998) elaborated in the NIE contribution.
In Loasby's thinking (1999), the rapid evolution in global economies is based on the growth of scientific knowledge and technology. The WTO treaty is an important catalyst of the scientific revolution. In the NIE terminology, each of institutions has its unique cognitive history. This is the main reason why the global economy under the WTO treaty is still a mosaic. The -rules of the game‖ are nation-specific or even clan-specific and sometimes reason-specific. In order to determine whether or not the transaction should be brought into the firm and to be taken out of the market, a firm should determine the frequency of the transaction under the conditions of uncertainty (Knight, 1920). When the uncertainty is high, vertical integration into an organization will be the most efficient mode of governance. This is the way how MNCs organize their global businesses.
The orthodox economics rely on competition to induce entry into the industry, since an entry of new firms provides the equilibrating function in the market to restore prices and profits to the competitive level. For the NIE, the most efficient firms are those that are able to enter a market and engage in the most specific transactions. In global markets, for example, a R&D-project or a cross-licensing is a complex chain of transactions that is not possible to implement without a strong contractual capability. MNCs are the most powerful actors of corporate capitalism (Schumpeter, 1942), not only because of production efficiency but because of their excellence in global contracting (Kumar, 1990(Kumar, , 1994(Kumar, , 1998. Through their strategies, MNCs use to restructure their markets and the global rules of the game. Because of the scientific technology (knowledge) revolution in the globalized economies, the neoclassical theories of the firm do not function properly. As Schumpeter understood, such a theory would have to deal with entrepreneurship and innovation in the process of technological revolution (Lazonick, 1991, p. 126) in which the NIE's notion of efficient contractual function is useful especially to start-ups and growth companies.
North (1993) claimed that economic institutions are path-dependent. Therefore, states can never be treated as exogenous actors in development policy, since states have the mandate or territorial monopoly to set property rights and to enforce competitive market conditions. Property rights matter, because they influence a firm's markets. The major part of costs and rewards associated with the use of intellectual property rights (IPRs) and, notably the use of IPRs does accrue to the firms. Fortunately, IPRs have the respectable connotation of property, instead of the unpleasant monopoly privilege (Machlup & Penrose, 1950). The Paris Convention for the protection of industrial property was established in 1883 and the Berne Convention for the protection of literary and artistic works was established in 1886, providing the internationally agreed frameworks for IPRs. The WTO TRIPS (Trade-Related Aspects of Intellectual Property Rights) agreement has established the standards of protection and the rules on enforcement, and brought the IPR regimes of the WTO member countries under the jurisdiction of the dispute settlement system of the WTO.
Knowledge is an important commercial commodity and the most profound characteristic of globalization. Universities are activated to the production of commercial knowledge for business purposes, following the University of California's role model. MNCs are actively involved with the universities and research labs in order to get access to the up-to-date scientific knowledge. The Mega-science themes, like the human genome project, are global in their basic nature since they can only be addressed on the global scale. The TRIPS is the main catalyst for the knowledge economy. Besides the TRIPS, the US innovation model is found to be the most -inspiring‖ institutional innovation globally (Haour, 2004;Loewenberg, 2009; Retrieved from www.economist.com/node/1476653).
The Bay-Dole Act (DBA), enacted in 1981, provides universities the first-right to commercialize patents, and, if they fail in that, the federal government retains the ownership of patents and then can grant non-exclusive licenses to interested third parties.
An institutional innovation was the liberalization of the federal antitrust laws to allow collaborating research projects of firms in the same industry: Stevenson-Wydler Technology Innovation Act of 1980. Ricardo (1817) is the father of the inter-industry trade theory. He was the first classical economist who developed the model of resource-based comparative advantage in international trade over regions. Ricardo used the opportunity cost theory. A country should specialize in production of that product (or service) whose cost for failure to produce is higher than that of the second alternative. A country would produce more of the two products than the other country (absolute advantage) or specialize in producing that product in which it has an advantage in utilization of the available resources (comparative advantage). Ricardo's concept of comparative advantage can be interpreted as a firm level competitive advantage of a firm's resources since a combination of firm-specific and country-specific resources is the means for the positioning in markets.

New Trade Theory
The Heckscher-Ohlin model (H-O model) (Ohlin, 1933) is built on Ricardo's model by predicting patterns of trade and production based on the factor endowments of a trading region. The model predicts trade patterns based on relative factor advantages. A country with a relative abundance of one factor is expected to produce goods that require a relatively large amount of that factor in their production. The original H-O model is an alternative explanation for free trade to Ricardo's model, rather than a complementary one. Ohlin understood the importance of the economies of scale that are external to the firm but internal to the industry or the region. Examples are skilled labor force and technological know-how. When economies of scale are significant, countries with few differences in technology and in production costs will gain from specialization in intra-industry trade in which countries trade similar, but differentiated, products with each other, e.g. different marks of cars.
The H-O-model is not generally accepted as a theory of trade. Empirical research has produced a number of results that do not match the logic of the model. Leontief (1986) tested the model in the US and found that a capital-abundant country exported labor-intensive and imported capital-intensive commodities. However, the problem of the so-called Leontief's paradox is not the H-O theory as such, but perhaps the concept of capital of orthodox economists. The US has its advantage in highly skilled labor. This can be seen as the intellectual capital. Using this definition, the exports of the US are human capital abundant, not particularly unskilled labor abundant. During the wartime, the US economy oriented towards innovations and human capital, which is the historical reason for the Leontief's paradox.
Linder tried to solve the Leontief's paradox, which questioned the empirical validity of the H-O-model, but did not solve it. Linder hypothesized (1961): Demand plays a more important role than comparative advantage in trade. In his model, a pattern of trade is determined by the aggregated preferences for goods within a country. Demand-based international trade arises from consumers' taste of variety. Countries with similar preferences are expected to have the same structures of industries. Aggregated tastes for certain product can emerge simultaneously worldwide which provides huge prospects for certain commodities (e.g. Nokia's mobile phones in the early 90s). Linder's demand-driven explanation is a complement of factor-driven explanations, because the supply side has dominated international trade theories.
Intra-industry trade means an increasing dependency on foreign markets (Foster & Stehrer, 2013), when industries are able to source intermediates from different countries, an activity referred to -offshoring‖. Intra-industry trade has evolved as an important economic practice increasing the number of differentiated versions of the same type products in markets, which is beneficial to firms and consumers, and gives businesses chance to benefit from the economies of scale and as well as use their comparative advantages. Intra-industry trade stimulates innovation in industry (Krugman, Obstfeld, & Melitz, 2012). The main benefit of intra-industry trade can be explained by using a car trade between Japan and Germany. Japanese cars are mainly family cars (e.g. Toyota), and German cars (e.g. Audi or BMV) are sport cars. Japanese and German car manufacturers can increase their global competitiveness by collaborating in non-core subcontracting and risk-sharing of radical technology, e.g., fuel cells.
According to Chamberlin, there are a lot of monopoly elements in international trade. Monopolistic competition theory of trade has been developed by Dixit and Stiglitz (1977), who analysed increasing returns and provided the foundations for the theoretical framework for combined analysing of economies of scale and product differentiation. The resulting interaction between demand for product variety and economies of scale leads to intra-industry trade (Helpman & Krugman, 1985). To Krugman, increasing returns is the fundamental cause of international trade and of comparative advantage of internationally acting firms, but its role has earlier been neglected because of problems of modeling market structures. The new trade theory (Krugman, 1979(Krugman, , 1980(Krugman, , 1981 is the standard in analysis of economies of scale and product differentiation. To provide consumers more quality, a firm under monopolistic competition maintains spare capacities of production, marketing and R&D in the same way as an oligopoly firm. This is risk-taking for the quality of consumption (Krugman et al., 2012). The monopolistic competition model does not predict in which country firms locate, but a comparative advantage in producing the differentiated good will likely cause a country to export more of that good than it imports. Krugman placed geographical analysis in the economic mainstream. In his theory, transportation costs are the key features of the home market effect. A country with the large demand for a good shall, at equilibrium, produce a more than proportionate share of that good, and be a net exporter of it (ceteris paribus). Instead of spreading out evenly around the world, production will tend to concentrate in a few countries or regions, which will become densely populated but have higher levels of income. This is a feature in Krugman's new economic geography (1991) in terms of the Marshallian triad (1995).

German Success Recipe by German HCs
Chamberlin coined the term product differentiation to describe how a supplier is able to charge a greater amount for a product than perfect competition allows. Product differentiation means the process by which sellers distinguish their products from others by making them attractive to buyers. German HCs apply their own models of market leadership in which Schumpeter's innovation/entrepreneurship concept and Chamberlin's product differentiation concept is combined. The term HC was coined by Professor Simon (1990, p. 876) who was the first to use it as a title of publication in a scientific German management journal. The first English book was edited by Simon (1996) in which the question discussed was: How Germany, the relatively small EU country in comparison to the US and China, has sustained its position as the number one exporter for many years?
HCs are normally known only in their own area, by customers and suppliers, but not to the wider public (Simon, 2009, p. 15). These highly successful firms are often concealed behind a curtain of invisibility and business secrets. Often, but not always, they are family-owned. Their values are conservative: hard work, high performance, and high employee loyalty. Because HCs are often locating in small towns, their main deficit might be competition of young talents worldwide (Venohr & Meyer, 2007. The main element of success by German HCs is a strong leadership with ambitious goals among the top three in global markets or to the number one on a own home continent. HCs produce high quality products that are ranked top in the world. For them, market leadership means -inner flame‖ to seek top performance, by unique products utilizing the in-depth knowledge about customers (Simon & Jonason, 2013, p. 144). HCs accept the risk of being a single product manufacturer. That is the reason why they need to grow and operate close to their customers. Customers are important drivers for innovations. HCs -earn‖ their market leadership through performance and not through price aggression.
Knowledge has been differentiated in terms of explicit vs. tacit knowledge (Nonaka & Takeuchi, 1995). Tacit context-specific knowledge, is difficult to create, transfer, or integrate via markets and, provides a rationale especially for German HCs to continue their amazing success-stories. Their tacit knowledge is valuable and provides competitive advantages, since it is less imitable, thanks to integrated marketing channels. The knowledge-creating theory focuses on the transformation and communication of what is already known tacitly by employees. HCs are excellent in that. Their strategic management system is auctorial, but their operative management system is participative generating superior performance embedded as -core competencies‖ within the firm (Hamel & Prahalad, 1994). Such specialized resources are developed, not acquired, and should have low mobility as Venohr and Meyer (2007), Simon (2009 and Simon and Jonason (2013) have widely reported.
HCs have their advantage organizational learning and technological innovations (Simon, 2009). HCs have efficient contracts (social, legal) between owner-managers and employees with -different personal utility functions‖ (Gutenberg, 1951). Simon's writings (2009Simon's writings ( , 2014 on German HCs revolutionize the US-dominating business theories and practices. Schumpeter (1934Schumpeter ( , 1939 illuminated the difficulties that a company has in its efforts to combine market-driven business processes and continuous radical innovations. HCs are doing that. They have occupied global leadership positions despite their small size. In more general terms, the greatest innovations are likely to occur from the cross-fertilization of ideas and professions. This is how HCs are acting. They are highly Schumpeterian in their action. The HCs' business recipe is working well in international markets. HCs invest in internationalization early in their growth paths. The high co-dependence between HCs and their customers means also a high risk to their customers (Robinson, 1933). In spite of leadership, HCs are not oligopoly companies. In oligopoly, there are significant barriers to entry and, thereby, a monopoly power. A high level of co-dependence between HCs and their customer is not an entry barrier, since the customers have a free choice. HCs are -hidden‖ but they do small things better than their competitors meaning differential advantage (Alderson, 1957(Alderson, , 1965. A market leadership position of a firm is highly rewarding but difficult to achieve in competitive globalized markets. In Drucker's analysis (1985) of entrepreneurial strategies, striving for leadership is the entrepreneurial strategy par excellence. This strategy is the core content of German HCs. Drucker's warning is that this strategy is the greatest gamble, making no allowances for mistakes and permitting no second chance. If it is successful, it is highly rewarding to a company. In average, the most rewarding entrepreneurial strategy is creative imitation-it is 90% of the whole. German HCs are the population of over 1,300 highly successful mid-sized firms. In Drucker's thinking, they are all gambling to achieve global market leadership. Their success rate as a population is near 90% (Simon, 2009) that is not possible in gambling in average. Following their integrating model of marketing, HCs use to develop their own unique resource configuration models that are oriented toward customer needs. The key issue is the humble choice of markets segments, to make good business in any kind of goods and articles, not to follow trends or hit lists.
The renaissance of monopolistic competition explains the survival of small firms in the global markets. HCs are best in that. They master the growth theory of customer-oriented firms. The empirical facts are unbelievable. Venohr (2010) has found that there are 1,500 firms in Germany that are world-market leaders (among three best) in their own segments. About 1,350 of them are HCs. About 90% of HCs act in B2B-markets and the most important industry group is the Machinery & Equipment industry. HCs prefer to specialize in globally heterogeneous and marginal market segments that MNCs use to avoid, because of low growth prospects and high transaction costs. MNCs attempt to establish their market powers through strategic alliances in-house R&D and portfolio investments abroad (Cross, 2000). HCs are mainly family firms having a strategic vision guiding marketing: to be the market leaders in their target markets. HCs are humble in constructing their marketing action plans, so that they are responsive to key customers (and competitors), in terms of Day (1990Day ( , 1992. The best success recipes are simple. German and Finnish marketing professionals are both honest and serious. German marketers have succeeded to commercialize their reliability, credibility, and authenticity. For Finnish marketers, the same kind of qualities means a handicap in international arenas. Finland has excellent MNCs that are at the same level as German MNCs. The major difference is SMEs. HCs have attained a high customer loyalty worldwide with their technological superiority. Finnish SMEs in B2B technology industries are in serious crises and only some SMEs in global digital businesses (e.g. Rovio) have succeeded. German HCs have excellent marketing skills. They understand completely the model of monopolistic competition that is the theoretical foundation of marketing by Chamberlin. Gutenberg's genius monopolistic scope concept advises German marketers to avoid oligopolistic power games that have been devastating such a big firm as Nokia. The strategic management model ( Figure 6) is constructed reflecting the German HCs way of acting, comparing to Abell (1980). The customer-focused marketing concepts, such as segmentation, positioning and the product-life cycle, are ingredients of strategic marketing. These may involve positions based on price, premium quality, superior service, and innovativeness. HCs have verified the doctrine of continuous product/service innovations in -old‖ technology industries (mainly -machinenbau‖). HCs have revitalized these industries by moving the competition logic away from price competition that dominates marketing of SMEs in -old‖ technology industries in most of EU-27-countries. German success recipe has been surprising, since monopolistically competitive industries are thought to be consumer service industries or retailing. One of the main reasons for Germany's competitiveness is excellent education (Schuman & Himmelreich, 2011). The German marketing doctrine is based on -Globalia‖, global market orientation (Simon, 2014). The market leadership doctrine has been known by any businessman worldwide, since the 1970s when the Boston Consulting Group's (BCG) experience curve (1970) in Figure 7 was published. But successful applications are few. The BCG's original claim for the experience curve is: For each cumulative doubling of experience, total costs would decline roughly 10% to 15%, because of economies of scale and learning-by-doing (Taylor, 1911), and that experience-based cost reduction can continue indefinitely. The practical contribution by BCG (1970) has been important. For the first time, there was a simple and parsimonious account of what competitive advantage is like, and how it is gained in the long run. The major difference between the BCG's experience curve and the HCs' global leadership recipe concerns pricing. HCs are not willing to win market shares by price wars. For HCs market, prices are the major market strategy element as Simon and Jonason (2013) elaborated. Their pricing strategy relies on price rigidity and product differentiation. This means a long-run commitment to serve customers and invent better products and services for them. Marketing channel is the key element of German success receipt. German companies prefer to internalize their marketing channels to keep customer secrets in a strict control.
Germany's customer-specific differentiation is not well known, since -global gurus‖ dominates the English literature and media. The paradox is that German firms have made a better version of the U.S. industrial method that helped the U.S. succeed for about hundred years until the 1980s. Sloan (1963), the famous CEO of GM, utilized Chamberlin's product differentiation in positioning. The U.S. management method is based on oligopoly models. The U.S. oligopoly theories are accountable to the mutual dependences between gigantic multinationals that are positioned in the mass-customized products and services and try to dominate core market segments by internalizing them. The majority of multinationals follow the U.S. oligopoly method, elaborated by Porter (1980Porter ( , 1985. German HCs have top positions (Figure 8).

Conclusions
Germany is the successful story in international trade in relation to the size of nation (e.g. population). In Germany, there are near 400,000 internationalized SMEs (Mittelstand) and about 100,000 of them have had FDI-operations (Venohr & Meyer, 2007). Among big EU countries, Germany is dominating (Sinn, 2012) and France and Italy have serious difficulties to understand why German MNCs such as Volkswagen or Daimler are so excellent. German writers of international trade are not known worldwide. The US universities, e.g. Harvard and Chicago, fully dominate the international debate on the competition models. In the last few decades, the pendulum has swung back towards the German historical school of economics. Instead of relying on abstract neoclassical theories of international trade, the new trade theory has become the standard in analysis of economies of scale and product differentiation. Gutenberg's general management doctrine (1951Gutenberg's general management doctrine ( , 1955Gutenberg's general management doctrine ( , 1969 is relevant to researches of Albach (1980) and Schmidt (2000), since the proponents of Chicago hide behind the curtains moral hazards such as the US financial crisis in the end of 2000s (Krugman, 2009(Krugman, , 2010. The growth of MNCs in number has been remarkable rapid. In 2012, there were about 100,000 MNCs in the world and 900,000 foreign affiliates with total assets value about $57,000 billion and sales about $33,000 billion. MNCs as a firm group account for about 25% of the world GDP and their total share of the interregional trade of all commodities may be at the level of 80% (Retrieved from http://www.wto.org/english/news_e/pres13_e/pr688_e.htm). Including the intra-industry in which German HCs are world-champions, the distribution of trade is well in balance in Germany. Germany is exceptional, since the internationalization of medium-sized firms has been slow in most of the WTO-member countries. This is the reason why it is important to study carefully the global mobility barriers of SMEs. The high market turbulence in global markets provides a lot of market niches for medium-sized firms to conquer. This is an excellent foundation for their business strategies (Clifford & Cavanagh, 1985). The case Germany is the best contemporary example of researches of Schuman and Himmelreich (2011). A successful recipe of the US, Germany, and China is to keep a balance between multinationals and the diversified SMEs (Figure 9).
Start-ups and growth firms ME contribute strongly to job creation in all continents. They are important because of the hollowing-out process of the big industrial firms Hamel (1991) has noticed in the US. This means a global reallocation of activities and can lead to the loss of skills and competitiveness of the Western economy. Hamel believed that a short-term strategic horizon may mislead firms to cut down their jobs to better their competitiveness on the expense of developing the people-embodied skills for the long-term leadership. The global economy is challenging. Substituting labor with capital and technology, along with shifting production to lower-cost regions has resulted in huge waves of firm downsizings in the EU, the US, and Japan (Baily, Bartelsman, & Haltiwanger, 1996). The relocation of industrial activities out of the home-base is also called Wintelism (Hart & Kim, 2002). The importance of mastering the manufacturing (Ramaswamy & Rowthorn, 2001) is not perceived as the element of a nation's competitiveness. The formerly critical skills of the regional clusters may become commodities, when MNCs relocate their manufacturing worldwide. In small and open countries like Finland, this kind of misinterpretation of the IO theories may lead towards the economic disaster.
HCs apply their own models of market leadership in which Schumpeter's innovation/entrepreneurship concept and Chamberlin's/Gutenberg's product differentiation concept are combined as Venohr and Meyer (2007), Simon (2009, and Simon and Jonason (2013) have widely reported. With about 90% success rate, HCs are competing away their main competitors in the various market niches. Their behavior is certainly different from the dominant oligopoly behavior. It is possible to imply that they follow the key principles of monopolistic competition that Krugman (1979Krugman ( , 1980Krugman ( , 1981 has elaborated in terms of international intra-industry trade. HCs are also theoretically -hidden‖ since most of monopolistically competing industries are consumer industries and the market scope is national or Pan-American or Pan-European. There is a research gap both in theoretical and empirical research concerning intra-industry trade and B2B-industries. The major reason for that is that these industries are extremely heterogeneous and the statistical criteria are different in different countries. It is time-consuming to collect up-to-date data-bases of that sort of businesses, especially in international context. In order to testify the main hypothesis, authors need proper data-bases.
The theoretical-methodological reason is dominance of mathematics in economics. The revolution of game theory, e.g. Nash equilibrium concept (Shoham & Leyton-Brown, 2009), together with econometrics as the main method in empirical analyses led to highly complex empirical models of technological changes, merger analyses, entry-exit models, and market power analysis. The influential economists of the Chicago School have not been fascinated to analyze the real completive circumstances in markets, since their worries were more excessively low profitability under the circumstances of perfect competition, when only normal profit is available. The Chicago economists were hostile to the US antitrust laws, although their major problem was and still is the abstract modeling that odds real market conditions. They succeeded to decline the comparative merits of Chamberlin (Archibald, 1961(Archibald, , 1963. The theory of monopolistic competition was seen as a departure from the ideal of perfect competition and Pareto optimality, e.g. Kaldor (1983). Stigler (1946Stigler ( , 1968Stigler ( , 1976Stigler ( , 1978Stigler ( , 1983) urged economists to use the orthodox price-theory. Friedman (1953) extolled the virtues of a free market system with minimal intervention: -The market is always right‖ (Ebenstein, 2007) or -Market exchange is an ethic in itself, capable of acting as a guide for all human action‖ (Harvey, 2005). Galbraith's view (1956Galbraith's view ( , 1958Galbraith's view ( , 1967Galbraith's view ( , 1973 of post-war capitalism is challenging the Chicago economists. According to Galbraith's critics, modern capitalism is dominated by large firms that invest heavily in manipulative advertising to create artificial wants and barriers. His concept of countervailing power is a parallel concept to Schumpeter's notion of trustified capitalism. Countervailing power describes the level of collusion between large firms and the government (Galbraight, 1956(Galbraight, , 1958(Galbraight, , 1967. Galbraith (1973) found that the static economic efficiency is a major barrier to innovate. Large firms grow because of technological imperative. Their size owes to economies of scale, large R&D budgets, and the unique ability to incorporate new technologies. The paradox is that the scientist Friedman supported the idea of perfect market contestability, but the advisor of the US president Reagan said that his view was simply -money talks‖. The debate between the Harvard SCP and the Chicago School occurred largely within these guidelines (Hovenkamp, 2009). A paradox is that Porter (1980Porter ( , 1985 became world-known by replicating Chamberlin's ideas such as the concept of differentiation without giving credits to Chamberlin. In reality, monopolistic markets are the most common in international trade. A realistic pricing strategy is important for internationally operating firms of which German corporate sector is an excellent example. As Simon (1990Simon ( , 1996Simon ( , 2009) and Venohr andMeyer (2007, 2009) have noticed, German HCs that are mainly operating in B2B-industries are unbelievable success stories in global markets. In Germany, the theories of competition are based on the nations' own scientists. In most of neoliberal countries, there are profound crisis in competition theories (Colander, 2008). The SCP was discounted by game theorists mainly from the University of Chicago. In the 1980s, the SCP was taken over by game theoreticians in the lead of Porter, many universities suddenly stopped doing empirical research under the SCP and relied on Porter (1980Porter ( , 1985 that are simplifications of the SCP and intended for practical business managers. However, the SCP provided a well-verified framework making for empirical and statistical analyses of how firms are really behaving. This was one of the arguments why the SCP was the key instrument to implement the anti-trust legislation in the US from the 1950s to the 1970s.
Because of the Chicago campaign against the SCP, most of economists rely on perfect competition and on econometrics as the main empirical research method. As Krugman (2009Krugman ( , 2010 has explained that the global financial crisis is the result of the anti-competitive behavior of financial institutions in the US. This kind of huge market failure is the most striking evidence on a market failure and on the fact that the US regulatory agencies should still apply the logic of the SCP analysis a natural monopoly with an optimal size and the lowest possible cost is not inefficient because of public ownership as the Chicago cadre uses to claim. This kind of realistic policy is well implemented in Germany where the current and earlier state-owned firms contribute strongly to Germany's national economy and have succeeded extremely well in the global markets. In most of the developing countries, natural monopolies may behave as real monopolies that can be one of the major barriers in the growth of local entrepreneurship (Lahti, 2010(Lahti, , 2011(Lahti, , 2012.


extfieldsofstudy:
Physics
Computer Science
Medicine

s2fieldsofstudy:
Economics
Computer Science

text:
Competing Opinions and Stubborness: Connecting Models to Data

We introduce a contagion-like model for competing opinions that includes dynamic resistance to alternative opinions. We show that this model can describe candidate vote distributions, spatial vote correlations, and slow opinion consensus with sensible parameter values. This may suggest that all these distinct behaviors, previously understood using distinct models, may be different aspects of a more unified model of human behavior introduced in this paper.

Firstly, while different aspects of opinion dynamics can be described by a wide range of quantitative models, the driving features of the global dynamics seen in empirical data remain unidentified because models with significantly different local dynamics can explain seemingly related global patterns [12,22] and because different types of models focus on different global patterns [4,12,[21][22][23].
Secondly, individuals in opinion dynamic models often reach a state of complete consensus (in which all individuals holds the same opinion) [5,11,24,25]. Depending on how model timesteps are defined, this is in disagreement with empirical observations, such as the slow dynamics of language diffusion and coexistence of multiple dialects [21] or the slow dynamics and coexistence patterns seen in the spread of religions [19].
We mitigate the previous limitations with a new model, which we call the Competing Contagions with Individual Stubbornness (CCIS) model. Our model includes features of, but is fundamentally distinct from, the Susceptible-Infectious-Susceptible (SIS) model wellknown in epidemiology (for a review, see [26]), where * keith@umd.edu † wrand@umd.edu ‡ girvan@umd.edu individuals can be "infected" (persuaded) by their neighbors and recover to a "susceptible" state (have no opinion).
The CCIS model assumes contagion-like dynamics, because rumors and opinions seem to spread virally [27][28][29][30], meaning the cumulative probability of adopting an idea increases with the number of exposures to that idea; therefore a realistic opinion model should have "viral" dynamics.
Next, unlike, for example, the influential voter model (VM), in which at each timestep an individual adopts the opinion of a randomly chosen neighbor, and similar opinion models [4,9,10,23], our model contains an unopinionated state that nodes recover to, in order to account for a stable fraction of unaffiliated voters observed in polls [31]. Also, a reduction in poll volatility has been observed before an election [32] which we might interpret as a reduction in the recovery rate as individuals increasingly need to make a stable decision before they go to the polls.
The key distinctions between our model and the traditional SIS model are: 1. Over time, individuals can become "infected" with different competing opinions or "strains". However, at any given timestep, each individual can only hold one opinion. The SIS model, in comparison, considers the spread of a single strain through the population.
2. Ideas are spread outward from individuals who have "caught" a new idea, while the SIS model assumes infections spread "inward" from a neighbor, or neutrally between two pairs of neighboring nodes.
3. Individuals become less likely to change their opinion the longer they hold it, while the SIS model has no equivalent dependence.
The CCIS model deviates from the SIS model for the following reasons: Recently, epidemic and epidemic-like models have incorporated multiple strains and multiple infections, which breaks from the traditional SIS model [33][34][35][36][37].
Similarly, it is typical in opinion dynamics research to study the interaction of multiple opinions, alike to the interaction of multiple infection strains, because rarely do opinions completely coincide (e.g. who should be president). It is therefore natural for an opinion model to contain multiple competing opinions, rather than a single homogeneous idea.
The spread of epidemics is usually assumed to be inward (i.e. an individual is picked at random, and diseases spread from infected neighbors to the ego) or neutral (a link between individuals is picked at random, and the disease similarly spreads between neighbors) [38,39]. Here, we model opinions spreading outward from the ego to its neighbors. The difference between these possibilities is subtle and not necessarily perceptible when social networks are k-regular, meaning everyone has the same number of friends. When the number of friends is highly heterogeneous, however, high degree and low degree nodes vary in the ease at which they change opinion depending on the way in which opinions spread (e.g. see [5] for variants of the VM). The outward spread hypothesis has been used to find agreement between an opinion model and election data [12], so we assume this behavior in our current model, although all the qualitative features in the current model are present in all three contagion methods (inward, outward, and neutral infection processes). Furthermore, this scenario is more plausible when we imagine that individuals actively spread ideas, instead of passively take ideas in.
Finally, we allow for individuals to increasingly resist adopting a new opinion the longer they keep their current one, by hypothesizing that individuals have "stubbornness", meaning that the probability of adopting an alternate opinion decreases the longer the current opinion is held. Evidence for this behavior is observed when studying the effects of pre-trial publicity (PTP) on juror outcomes [40]. A meta-analytical review found that a delay between the PTP exposure and the verdict, especially after one week, produced a significantly greater bias, suggesting that stubbornness towards an initial opinion increases with time, at least on the timescale of a week. Interestingly, stubbornness is not unlike the primacy effect, well studied in psychology [41,42], where the first idea someone hears can be the idea someone favors regardless of its validity. That effect, however, usually concerns ordering of choices, and not when each choice was offered.
In the following sections, we show agreement between our model and several empirical studies. Firstly, the model is shown to describe the collapse in vote distribution among candidates when votes are scaled by Q/N , where Q is the number of candidates and N , the number of voters [12,13]. Secondly, we argue that the CCIS model approaches the voter model universality class (VMUC) in certain limits [43], and therefore diffusively spreads voter preferences. The spatial vote correlations in this limit match those observed in several countries [22,44]. Lastly, we show that model parameters allow for opinion consensus to be reached at arbitrarily long times.
The remainder of the paper is structured as follows. We begin by describing past research (Section II) and details of our model and algorithm implementation (Section III), before comparing our model to empirical data (Section IV). Next, we use simulations and analytical calculations to show how our model can vary between fast and slow consensus (Section V). Finally, we conclude our paper (Section VI).

II. BACKGROUND
In this section, we review empirical data that motivates our model and recent models that, like our CCIS model, exhibit slow or non-consensus.
In recent years, Big Data has allowed researchers to observe large numbers of social interactions [12,13,22,27,30,32,45,46], leading to new insights into opinion dynamics. Here, we focus on two themes which have received recent attention: candidate vote distributions [12,13,22] and spatial vote correlations [22].
Vote-share distributions for candidates in elections show relatively universal properties. For example, two recent studies demonstrated that the vote-share distributions, when scaled by Q/N , often collapses to a universal function, based on election data from several countries across at least 50 years [12,13].
Other recent work has shown that the vote-share correlation in the US elections and the turnout rate correlation in European elections decreases as log(r), where r is the absolute distance between two voting districts [22,44], meaning correlations are very long-range. Interestingly, this is a prediction of the VM at an arbitrary, fixed time [9,10,47]. In the continuum limit of the VM, the two point correlation function in two dimensions decreases as ∼ log(r)/log(t), where t is time.
Opinion dynamics research has recently focused on how to slow or stop the consensus of opinions in models [2,4,6,14,15,23].
For example, the Non-Consensus Opinion (NCO) model assumes that individuals asynchronously update their state to be that of the majority of their neighbors plus themselves (if there is no majority, a node does not change) [4,23]. This rule, a variant of which was originally created for a cellular automaton [48], seems to converge to a frozen state of non-consensus, when each opinion initially has a roughly equal share of the population.
Several models similarly achieve non-consensus while, in addition, populations of a given opinion vary in time. For example, in a model by Galam [2], some individuals are contrarians, who oppose the majority opinion in their local neighborhood, while all others follow the local majority. Galam showed that contrarians create a non-consensus of opinions, and that there is a critical threshold in the fraction of contrarians, above which opinions are, in equilibrium, equally divided within the population. A few authors also explored the idea of adding "stubborns", which never change their opinion, but can change the opinion of their neighbors, to the Galam model [14], and to the VM [6,15]. In both cases, the equilibrium distribution is not consensus.
Lastly, we mention recent work on contagion and contagion-like models with multiple strains. A few recent studies have noted that the coexistence of two contagion strains on a single network is unstable [36,49], which is in agreement with the work presented here, although they seemed to have focused on when β 1 /δ 1 = β 2 /δ 2 , where β i and δ i are the strain-specific infection and recovery rates, respectively. Our work, in contrast, (Eq. 7) suggests that even as β 1 → β 2 = β and as δ 1 → δ 2 = δ → 0, we similarly find that two-strain dynamics is unstable because we approach the VMUC, implying finite expectation times for two strains to coexist on a finite network. Initial simulations show that the coexistence time decreases with δ and approaches a minimum, as δ → δ c , the critical point after which all strains die out exponentially fast. We have therefore taken a step towards showing that twostrain dynamics is always unstable on a single network. Theoretical (versus numerical) support for δ > 0 is still needed, however.
Methods to create non-consensus with contagion models often employ multi-layered networks where each strain, disease, or opinion exists on its own network, which is then coupled to one another [34,36,37]. Typically, these networks have distinct topologies, or, when the coupled networks are identical, certain parameter combinations are found to achieve non-consensus [37]. Alternatives include dynamics on a growing network [50], or coupled populations that are individually mixed [35].
The CCIS model is distinct from the above models in that every node holds onto their most recent opinion with increasingly stubbornness, but can return to the unopinionated state at random (see Fig. 1), and all dynamics are on a single network. Table I defines the different parameters and variables of our model: β parameterizes how quickly individuals change their opinion (i.e. their "persuasiveness"), where individuals change their opinion with a probability proportional to β. Next, µ parameterizes the speed at which individuals become stubborn: at a linear rate, individuals are decreasingly likely to change their opinion until they have kept an opinion for µ −1 timesteps, after which an individual's opinion is temporarily "frozen". Finally, δ is the rate at which individuals "recover" from an opinionated to an unopinionated state, and are thus susceptible to a new opinion again.
Our model's increasing stubbornness property was also explored in the VM with the additional rule that after a certain period, individuals could change their opinion at a fixed rate [25]. It was found that stubbornness i ncreases the speed of consensus, and for a non-trivial value of µ, it can maximize the consensus speed. We find a similar behavior in the CCIS model (Fig. 8), although our model is most alike to the VM with a freezing period, previously explored in one dimension [11], because it is easy to argue that for a period ∼ (δ −1 − µ −1 ), an individual's opinion is frozen. This model too exhibits the same qualitative behavior, suggesting it is a fairly generic property of decelerating microdynamic models. A related model, where individuals can change their opinion after a certain number of neighbors attempt to convince them [24], creates a much faster consensus on a grid graph than the VM and created curvature-driven coarsening, consistent with an effective surface tension, which the traditional VM lacks. Taken as a whole, it would seem that adding a stubbornness-like property would create exceedingly fast consensus, even though we notice that few competing opinions, e.g. opinions about operating systems or presidential candidates, achieve consensus after several years. The CCIS model can substantially increase the consensus time, however, in the limit that δ → 0. This limit is in agreement with empirical data (Fig. 2), and we claim that certain parameter combinations in this limit can produce VM-like consensus (Eq. 1). Furthermore, unlike the previous models, the CCIS model is contagion-like, which we've previously mentioned is in strong agreement with empirical data. Of course, one reason we assumed δ > 0 was to allow for a population of individuals without an opinion (i.e. are "independents"). In this paper, we will focus on when the effective number of independents is small. Opinion A density as a function of t and τ P A Fraction of individuals with opinion A at a given time

III. MODEL DETAILS
We assume in this paper that: 1. Interactions between individuals are localized on a fixed, unweighted, and undirected network topology.
2. Opinions do not evolve.
3. Some individuals begin with initial opinions and the remainder begin in the unopinionated state.
The fixed topology hypothesis is not necessarily a good approximation, because individuals tend to interact in a temporally varying network [51,52]. However, we expect the qualitative behavior to be the same in the case of slow network evolution, and the behavior in the fixed network case is analytically tractable. We believe an undirected network, however, is a good approximation for social interactions, because we expect a 2-way street between the influencer and influencee, and without a priori detailed knowledge of the real social network, we can reasonably assume the network is unweighted. In many practical instances, the assumption that opinions do not evolve works well, especially when the number of opinions possible is fixed (e.g. voting "guilty" or "not guilty" in a jury or picking among Q candidates). Again, this assumption also allows for analytic tractability, and minimizes extraneous variables.
Lastly, the assumption that individuals may have initial opinions is taken for granted in models like the VM, but is not guaranteed in many real world situations. For example, voters may start out without any knowledge of party candidates, although their understanding can increase as they interact with knowledgeable friends.
We now describe how we implement the CCIS model in an agent-based simulation. The model is run on a network of order N , where the state of each node, i, is η i ∈ {0, 1, 2, ..., Q}. Q is the total number of opinionated states, while 0 corresponds to no opinion. At t = 0, n 0 nodes are in state 0, n 1 are in state 1, ... , n Q are in state Q, such that n 0 + n 1 + ... + n Q = N .
The dynamics are as follows: 2. Count the number of opinionated individuals, N op = N − n 0 , and repeat from step 1 with time incremented by ∆t = (N op (1 + δ)) −1 .
In our model, τ is the "time a node has had its most recent opinion", µ is the stubbornness rate, β is the persuasiveness and δ parameterizes the probability of moving to the 0 state at random (see Fig. 1 for a diagram of the dynamics, and Table I).
While it may seem odd, at first, why ∆t = N −1 and the recovery parameter is not δ, the reasoning follows from the work of Boguna, Castellano, and Pastor-Satorras [39] for the SIS model. ∆t is normalized such that N op nodenode interactions take place, and δN op of the opinionated nodes recover, at time i ∆t i = 1. Basic algebra implies that, holding N op constant for each timestep ∆t = ((1 + δ)N op ) −1 . Similarly, this reasoning implies the recovery probability is δ/(1 + δ).

IV. AGREEMENT WITH DATA
We make two claims about the CCIS model's agreement with data: We can reproduce the scaled distribution of candidate votes observed previously in several countries [12,13], and the vote-shares correlations as a function of distance seen recently in presidential election data in the US [22].
Firstly, we discuss how the model reproduces the scaled distribution of candidate votes (Fig. 2). Three fitting parameters are used: degree distribution, persuasiveness (β = 0.5), and initial seeding (6% with an initial candidate preference). Individuals interact on a scale-free network with a degree distribution, p(k) ∼ k −α , where the scale-free nature of this graph is a natural construct because many social networks studied have scale-free distributions, p(k) ∼ k −α , where 2 < α < 3 [54]. It was found that α = 2.9 helps create close agreement to the distribution, although the coefficient did not strongly effect the fit to the large vote-share tail. Lastly, we let δ = 0 such that the distribution does not vary in equilibrium, when µ = 0.3.
We see, remarkably, that the model results almost lie on top of each set of data for vQ/N > 10 −1 . Of the elections modeled, we find that only Switzerland's diverges significantly from our model due to its unusual "doublehump" distribution, plausibly because votes are swayed by the local language differences (primarily French and German). We find, however, that other unusual features, such as the kink in the distribution tail of Poland's 2001 election, seem to be faithfully reproduced, suggesting it is not part of the "complexity of human nature and context-dependent factors" as initially assumed [13]. When vQ/N < 10 −1 , the model diverges from the data,  [13], where data is shifted down by decades for clarity (inset shows the original data collapse).
Here v corresponds to the number of votes, and Q and N were the same as the empirical data. Three fitting parameters were used: (1) power-law degree coefficient for each election network α = 2.9, (2) persuasiveness parameter β = 0.5, and (3) 6% initially seeded with a preference to a candidate. Two others were fixed: µ = 0.3, and δ = 0, where the latter assured non-consensus distributions in equilibrium a .
a Five elections out of 2109 were not modeled (one in Finland, 1995; one in Italy, 1958; and three in Italy, 1979) all with N < 10 2 because Q greater than 6% of the nodes. Although we could have included these with each candidate as a seed (the effect is negligible), we removed them to keep initial conditions consistent.  Figure 3: The correlation as a function of distance, with regions of 5 × 5 node squares (N = 10 6 ) after 8 × 10 3 timesteps. The top line is for a 2d grid with periodic boundary conditions while lower lines are for small-world networks built starting from a 2d grid in order of increasing fraction of edges randomly rewired. C(r) ∼ −log(r) even when a significant number of edges are re-wired, thus behavior is robust to graph "noise" symbolizing weak ties between distant neighbors [53].
which may suggest that, in this limit, a new set of dynamics comes into play. For example, candidates are who are extremely unpopular may drop out before an election, thus reducing the probability for those candidates to be on the ballot. Differences between country distributions are strongest in this limit as well, suggesting that country-dependent factors play come into play too. Our model is influenced by a previous one we call the Fortunato-Castellano (FC) model [12] which is known to approximately recreate rescaled vote distributions in Poland, Italy, and Finland. Alike to the FC model, the CCIS model considers opinions spreading outward, and individuals interacting on a scale-free network. We improve upon the previous model by adding agreement with spatial vote correlation data, slow consensus, and through simulating each election on a single social network. We describe the former two points in what follows, but for now, we focus on the final point. In the FC model, it was assumed that for every election, each of the Q candidates tries to "get out the vote" among Q disconnected social networks. Candidates however, seem to often compete for a common set of voters [55][56][57], therefore, it may be more reasonable to assume that the electoral network is connected. We can accomplish this in the CCIS model by taking δ → 0, which produces equilibrium vote distributions that seem to non-trivially collapse into a single function when votes are scaled by Q/N . Fundamentally, this is accomplished by the opinion stubbornness parameter forcing a non-consensus equilibrium, thus providing further evidence that stubbornness may exist in opinion formation.
Secondly, we show how the CCIS model creates C(r) ∼ −log(r), seen in empirical studies [22,44]. To do so, we must first argue that our model approaches the VMUC. We conjecture that the VMUC is approached in our model when [43]: where k and k 2 is the first and second moment of a network's degree distribution,respectively. The reasons are as follows, which we will expand upon in Section V: 1. δ β k 2 / k ensures the model is a 2-state system, like the VM (for a proof that this breaks down when δ β k 2 / k , see [38]).
3. δ µ is numerically found to increase the consensus time. therefore our model can reproduce the vote-share correlations as a function of distance, C(r), seen in empirical data [22,44] (see Fig. 3).
The relation above is self-consistent by setting Eq. 4 to 0 (a necessary condition for VM-like diffusive consensus), and assuming T cons , the mean time to reach consensus (Eq. 7), is 1.
Numerical studies with δ and µ > 0 have so far found that T cons log(N )/λ 1 , unlike the VM, where T cons ∼ N for a graph with bounded k 2 . If δ µ, consensus time is increased, which is equivalent to λ 1 → 0, a regime where we expect to approach diffusive consensus.
The argument that we approach the VMUC for a kregular network is as follows, given the conditions in Eq. 1: 1. After a time ∼ βkδµ −1 , a given node infects a neighbor.
(b) For a brief period µ −1 the node is susceptible.
(c) The probability of an opinion changing is always on the order of βk (d) Therefore the probability a node is susceptible times the probability a susceptible node changes its opinion is βkδµ −1 2. Nodes are updated at random. Therefore, the model acts like a VM with time rescaled by βkδµ −1 . This argument ignores the exact nature of p rec (δ) or p inf (β, µ), the time distributions to recover and change an opinion, respectively, and is therefore a general outcome of many variants of our model.
On a d-dimensional grid, where the degree of each node is 2d, the correlation as a function of time and distance in the continuous limit, is that of the VM, with time rescaled by 2βδµ −1 d [47]: where D = (2d 2 µ −1 δβ) −1 and a is the lattice spacing. The correlation approaches one exceedingly slowly for d = 2, and we find, in simulations where δ, µ → 0, that the spatial correlation can be effectively frozen in time (Fig. 3). To be consistent with previous work [22,44], we define the normalized correlation in our figures as: where P i is the fraction of voters for candidate 1 within a small region, P is the average fraction of voters for candidate 1, and σ 2 P is the variance in vote distribution across all regions. P i P j |d ij r is the 2-point correlation function between regions a distance r from each other.
We find that C(r) ∼ −log(r), even when up to 10% of edges are re-wired (Fig. 3), and that as more edges were rewired, the time-dependence on correlation falls significantly. The nearest-neighbor correlation does not change significantly for p ≤ 10 −2 , even as the mean distance falls by half. The robustness of correlation behavior as mean path length decreases is not unlike the robustness of graph clustering to rewiring seen in the Watts-Strogatz model [60].

V. APPROACHING EQUILIBRIUM
Complete consensus, where all nodes are in some state q > 0, is achieved for most parameter values, with the exception of β = 0, or δ = 0 and µ > 0. We will show that the time to consensus, however, can be arbitrarily large. But first we must first determine a realistic consensus time. It is not well defined what a timestep really is in the real world because we interact in a highly bursty manner [52,61] and the influence of a single interaction has the potential to be almost immeasurably small [30] or significant [27,29,45]. Nonetheless, if we assume adoption on the timescale of days to weeks, as has been assumed previously [2], we can search within the model's parameter space for consensus times of, for example ≥ 10 3 timesteps, given that a consensus among voters on who should be president is not achieved on the timescale of years [15].
In this section, we will begin by understanding a significant, but short-lived, transient our model creates, before we explore the consensus time in the most realistic scenario (i.e. where C(r) ∼ −log)(r)), where our model approaches the VMUC.
We begin our analytic treatment by studying the transient behavior of our model as δ → 0. We notice a transient "jump" in the fraction of individuals following a given opinion on a timescale much smaller than the time to reach consensus (see inset of Fig. 5). One way to analyze this behavior is via a mean field PDE equation, which we derive in Appendix A: In Eq. 4, ρ A is the density of individuals as a function of t (time) and τ (the time an individual has kept their most recent opinion). P A is defined as: i.e. the fraction of individuals with opinion A, while B are all contrary opinions. Finally, k is the average degree, and all other parameters are as defined previously. Cyan equation elements correspond to loss via recovery, and blue elements correspond to gain via convincing those with an alternative opinion. For simulations and analysis of the PDE, we set δ = 0, in order to observe nonconsensus equilibrium values seen in simulations (Fig.  4). We distinguish this theory from a completely distinct type of mean field theory to be discussed shortly, by calling it the PDE mean field theory (PDE MFT).
Using Appendix A and methods described in [25] for the VM with a similar stubbornness property, we find that it is unstable for everyone to have an equal vote share, at least for Q = 2. The initial condition diverges exponentially due to the largest eigenvalue being positive. If P A is the fraction of nodes with opinion A, then at initial times, |P 1 − P 2 |(t) ∼ Exp(λ 1 t), where, assuming µ 1: The derivation of this value is in Appendix A. We test this theory by showing it is effect on the timescale of the transient. A simple derivation shows that the transient time to the δ = 0 equilibrium is: where ν and c(µ) are fitting parameters. Fig. 5 demonstrates close agreement between theory and simulation. When βk = 1, this eigenvalue should agree exactly with the value cited previously [25], but we find disagreement by an overall pre-factor of 1/4 which, at least to our knowledge, may have been missed in the previous work. When δ = 0, we almost never reach consensus but in the more realistic case where δ > 0, we always reach consensus, although potentially at very long times. We argue in Section IV that we approach the VMUC as δ → 0 and µ β k , therefore, in this limit we can employ powerful theory on the VM [5] to find the scaling behavior of the mean consensus time: where, again, k 2 is the second moment of the degree distribution p(k) (see the Appendix B for a derivation of this result). To distinguish this theory from PDE MFT, we call it the VM-limit mean field theory (VML MFT). We explored the agreement with this theory by varying δ, µ, β, and k 2 , while separating the respective timescale sufficiently in order to understand the individual effects.
To explore the dependence of the VML MFT on k 2 , we set δ, µ → 0 such that δµ −1 = 1. It is easy to show that this limit is still captured by the equations, but the consensus times are found to be much faster than with δ > 0 and µ > 0, for appropriate (VM-like) values of  Figure 6: Mean consensus time, T cons , for scale-free networks (p(k) ∼ k −α ) with β = 0.5, δ = 0, and µ = 0. Inset is one example of consensus with P 1 (t) and P 2 (t). Using the VML MFT, the expected fit (solid lines) is Eq. 7. Simulations are averaged over 10 networks (30 networks for 3 × 10 4 ≤ N < 10 5 , and 20 networks for N = 1.2 × 10 5 ) with 100 trials per network.
each parameter. In a scale-free network (p(k) ∼ k −α ), k 2 diverges with network order, N : Therefore: Fig. 6 compares our simulations to the VML MFT. Although a finite size transient impedes this scaling behavior for N ≤ 10 4 , we still see agreement large enough networks. For Poisson networks, we see T cons ∼ k 2 −1 = ( k 2 − k ) −1 in the inset of Fig. 7.
Next, we focus on whether T cons ∼ β −2 , holding all other variables constant. Interestingly, agreement with theory was closest when β ∼ O(1) and k ∼ 10 − 20. The discrepancy between theory and simulation may be due to finite size effects, but future work is necessary to test this assumption.
Finally, we studied T cons by varying δ and µ. Plotting T cons versus δ (not shown), we find that T cons ∼ δ −2 for δ < β k 2 / k and µ ∼ 0.2, although this breaks down for small δ, presumably for the same reason that Eq. 7 breaks down for small β (Fig. 7). We also explored the regime where δ µ (e.g. µ = 0) and found the consensus and solid line (T cons ∼ k 2 −1 ) in the inset. varying δ. Theory (T cons ∼ µ 2 , cf. Eq. 7) seems to disagree with our simulations, at least in the parameter space observed (colored lines indicate best fit slopes, which are all around T cons ∼ µ 5 to within error bars).

Û Û Û Û Û Û Û Û Û Û Û ÛÛ ÛÛ Û Û Û Û Û Û Û Û Û Û Û Û Û ÛÛ Û Û Û Û Û Û
time dropped significantly when δ β k 2 / k , on networks with strongly bounded k 2 , suggesting that near the critical point for a long-lasting infection, β/δ = τ c , the consensus timescale is an order parameter that presumably drops to 0 in the thermodynamic limit. In comparison, timescales like the time to reach equilibrium blows up in this limit. Whether this can aid in determining the critical point for the SIS model for arbitrary graphs is as of yet unknown, although significant recent interest in determining the critical point for arbitrary graphs suggests this idea is worth exploring [38,39,[62][63][64].
The behavior of T cons versus µ, however, demonstrates a breakdown in our theory (Fig. 8).
Theory dictates that T cons ∼ µ 2 , for large enough µ, but instead T ∼ µ a , a ≈ 5 for the parameter values explored. Interestingly, there is also a non-trivial minimum in consensus times, as seen in similar VM-like models [11,24,25]. Lastly, there is a minimal consensus time, It should lastly be mentioned that when δ = 0, µ = 0, and βk = 1, the CCIS model is alike to the invasion process (IP) [5], where a neighbor is randomly chosen to have the same opinion as the root node [5]. In the true IP, T cons ∼ N k −1 k , but in the CCIS model, T cons ∼ N k 2 . The discrepancy is due to a fixed fraction of neighbors, 1/ k , being changed in the CCIS model, instead of exactly one in the IP. Interestingly, this implies that T cons ∼ 1/N in a complete graph. Initial simulations suggests this prediction is correct for large enough N . For moderate N , in contrast, the consensus time reaches a maximum.

VI. CONCLUSION
In conclusion, we have introduced a model of opinion spread that agrees with some current data on opinion dynamics. Furthermore, we have shown that the speed of consensus in our model can be made arbitrarily slow, in agreement with the expected long times for groups to reach consensus. Lastly, our model makes few assumptions, implying it can plausibly explain a range of behaviors mentioned in Section V.
Here, we discuss some qualitative conclusions that we can draw from the CCIS model, which future empirical investigations may be able to corroborate. Firstly, this model may qualitatively describe properties of colicin-containing bacteria. Colicins are a family of toxins produced by strains of certain bacterial species (such as Escherichia coli and Streptococcus mutans) that are active against competing strains but not their own. The stubbornness property of our model can be thought of as reducing the impact of an opposing strain on a single host, alike to colicins [65], because the build-up over time of colicins would make it less and less likely that an alternative strain can successfully take over the host. Alternative models of competing strains [34-37, 49, 50], however, assume coupled networks or a dynamic network, and furthermore, have no time-varying susceptibility. In contrast to these other models, the CCIS model suggests that colicin-containing bacteria should allow for greater genetic diversity than similar bacterial species lacking colicins, just as stubbornness in opinions allows for arbitrarily slow approaches to complete consensus. Furthermore, the relative success of a single strain (its share of the entire infected population) should follow a distribution similar to Fig. 2.
In addition, the model can be interpreted as the "viral" spread of competing products, where increasing stubbornness is instead increasing brand loyalty. With these reinterpretations, the CCIS model suggests that there is a greater number of products in the buyer network, compared to when individuals adopt a product readily, without stubbornness, or alternatively, "brand loyalty" [66,67], because the timescale for "consensus", i.e. where a single product dominates, can be extremely long. Furthermore, the brand-share distribution should exhibit a distribution similar to the one seen in Fig. 2.
Future work is also necessary to explore ways in which our model can be made more realistic. For example, it may be more appropriate in future work to add heterogeneous stubbornness, because some individuals can stubbornly refuse to relinquish an idea, while others may shift their stance like a chameleon. The most stubborn individuals can help make the political preference in a two party system near the 50/50 mark, as explored previously [6,14,15]. Expanding on previous work, we expect adding heterogeneous stubbornness to our model can slow down or stop consensus, depending on the heterogeneity, potentially creating better fits with current data.
It may also be more realistic to replace the recovery probability with a random opinion flipping probability. Adding random opinion flipping to the VM seems to have allowed for better agreement with empirical data in a recent paper [22], and, interestingly, this replacement in our model seems to create three critical points, between order (relative consensus) and chaos (no consensus). Trivially, if the flip rate is high, two opinions should be equally split. In comparison, when the flip rate is moderately low, one opinion should dominate. There seems to be a 1 st order transition between these two extremes for a high flip rate, and a continuous transition for a small but non-0 flip rate. Finally, a flip rate of 0 tends to force one opinion to dominate, unlike the behavior for an infinitesimal flip rate. Recall in our current paper, recovery always forces one opinion to dominate, regardless of how large the recovery probability is. Despite a relatively small change in the dynamics, its clear that the behavior is both interesting and drastically different.
In a similar vein, we might expect that real opinions exhibit a ladder-like process where individuals move between opinion extremes. This creates non-consensus in continuous opinion models [68] and thus may slow the consensus in a discrete model. These new dynamics may help us understand how polarization happens when individuals stubbornly hold onto an opinion.
As a final point, we only compute the theoretical consensus-time scaling in the mean field limit. More accurate results may be found with, for example, a simple pair approximation [69], or a heterogeneous pair approximation theory [70], which can also take into account degree assortativity.
Appendix A: Derivation of Eq. 5 In this appendix, we use the CCIS PDE MFT to understand the initial transient in the opinion densities (see inset in Fig. 5). This behavior has the potential to significantly effect the time it takes for our model to reach complete consensus, especially when the persuasiveness parameter (β) is of O(1) (Fig. 4).
The equation used to describe our model in the mean field is: (t) Figure 9: Schematic of the A th scalar variable in Eq. 4 as a function of time and length of time since the adoption of opinion A.
With the boundary conditions where B represents the opposing opinions and: Let us first understand the variables in our equation. The independent variable, t, is "time" and τ is the "time since an individual kept their most recent opinion". The dependent variable, ρ A (t, τ ), is the opinion density for opinion A (see Fig. 9).
The RHS describes the ability of individuals to recover (cyan term) as well as the ability to change opinions (blue term). We can interpret the boundary conditions as: 1. Normalizability 2. An increase in the infection density due to unopinionated neighbors (cyan term) and opinionated neighbors (blue term).

Initial conditions
See Fig. 10 for a visual representation. Despite the simplifications of the equation, we find good agreement with simulations (Fig. 4). We therefore have a simple equation which allows us to understand the reason for the transient.
Following [25], we set δ to 0 and let Q = 2 (creating a 2-state system), and the discretize τ to derive a set of equations that we linearize around a fixed point to determine the speed of the jump (Eq. 5 & 6). Due to the above assumptions, this is only accurate for µ 1. We define the following macroscopic variables: where τ is shorthand for ∞ τ =0 . If we let Ω(|) be the conditional probability function, andẋ ≡ d dt x, then Eq. 4 becomes (for τ > 0): and for τ = 0:  With an equivalent set of equations for ρ 2 (t, τ ) and (A4) From the above results we can sum ρ 1 (t, τ ) to find the equations for the macroscopic variables: To lowest order in µ, we also find that: I 1 (t) µ(1 − βk)P 1 (t) + βk[µP 1 (t) 2 + I 1 (t)P 1 (t) − I 1 (t)] (A6) These equations are solvable by expanding around the solution P 1 = P 2 = 1/2 and I 1 = I 2 = µ[1/(βk) − 1/2] to first order. The resulting largest eigenvalue is Eq. 5.
Appendix B: Derivation of Eq. 7 In this appendix we derive Eq. 7 using a VML MFT, which is distinct from the PDE MFT in Appendix A. Our derivation is heavily based on the derivation of consensus times for the VM and Invasive Process by Sood, Antal, and Redner [5]. We use the same conventions as in that paper, except the transition probability scaling factor (S), the degree distribution (p(k)) and associated moments, k m = k p(k)k m .
Let η(x) be the state of a node x on a network with adjacency matrix A xy and order N . Assuming 2 opinions and δ → 0, we have a two-state system. Using the conventions of Sood, Antal, and Redner [5] the opinions of the two-state system are "0" or "1" instead of "1" or "2". We stress that the 0 state is an opinionated state.
Lastly η x is the state of the system after changing a node x: η x (y) = η(y), y = x 1 − η(x), y = x (B1) The transition probability at node x is therefore: where, if we assume δ → 0: Effectively, with probability β an attempt is made to convince a neighbor but only on the order of µ −1 δ of the time is the node available to be convinced (and is otherwise stuck in its state), therefore, our model effectively scales the conviction rate by βδµ −1 .
We further assume a mean field solution, where the adjacency matrix becomes the average adjacency matrix.
Instead of individual states η(x), we can instead focus on ρ k , the density of states with degree k.
Here, x is the sum of all nodes with degree k. To clarify the below equations, we also define a variable ω: Next, we define our raising and lowering operators for ρ k , which defines the probability of increasing or decreasing ρ k by a small incriment: This is to say, the change in ρ k from a neighbor with average degree k nn = k 2 k . The raising operator is defined as: With simplification, this yields Similarly, for the lowering operator: It can be shown that, ξ 1 , the probability for all nodes to reach state one in equilibrium (also known as the "exit probability") is: And similarly, that ρ (or magnetization, if this were a spin system) is a conserved quantity. The time to consensus is: Here, the average timestep is the average number of interactions of a neighboring node with degree k nn We expand to second order in ∆ρ k and find that As was shown in in the original paper [5], this value reaches 0 for time T cons ∼ O(1) which is much smaller than the next term: Lastly, by a change in variables: Because, ρ is conserved and v k → 0, ω → ρ. Therefore, we find that this equation simplifies down to (24) in [5], where T cons ∼ N → N ef f and Thus, we have an effective scaling of our consensus time. There is a discrepancy between theory and simulation for small values of δ, β, and presumably µ, potentially because of the finite variance of p inf (β, µ) and p rec (δ), the time distributions to infect and recover, respectively, or possibly because of finite size effects. Future work could improve upon the current results by correcting for these effects. In addition, a pair approximation theory could provide correction terms to the mean field assumption [69,70].