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Large-scale risk events in the venture capital (VC) market can easily lead to systemic risk in the financial market. This paper collects the comprehensive dataset of Chinese VC market from 1999 to 2020, and constructs the multi-layer networks of VC market. After that, a risk contagion model of venture capital market is designed considering both the capital loss transmission and social relations effect (investor herd behavior). By simulating various situations with different risk origins and scales, the speed of risk contagion and the total impacts on the overall stability of the market (network) are compared. We find that the herd behavior of the market will generally aggravate the consequences of risk contagion. Compared with unicorns, the failure of a leading VC firm can cause a wider range of risk contagion. The contagion consequences caused by the failure of random financing companies are more serious than those caused by random VC firms. Moreover, we identify that telecommunications services and information technology are high-risk industries in VC market. Based on the empirical results, this article provides policy implications for the market regulatory sectors to prevent VC market risks. [ABSTRACT FROM AUTHOR]

Venture capital market is one of the most important financial markets, which plays an important role in promoting industrial innovation and economic development. In this study, the complete data set of all venture capital events occurred in China from 2009 to 2020 was used to construct two kinds of complex networks, whose topological property and dynamic evolution trend of the network are studied. Specifically, we construct the co-investment network among investor, as well as the binary complex network of investor-entrepreneur, and propose the evaluation method of the importance and performance of venture capital institutions. Based on the proposed method, we identify some leading venture capital institutions with high performance and importance. Furthermore, this paper enlightens venture capital practitioners by discovering the investment behavior and preference of these leading institutions. [ABSTRACT FROM AUTHOR]

SummaryWe look at the problem of coloring 1, 2, …, nwith rcolors to minimize the portion of monochromatic k-term arithmetic progressions. By using residues to color Zmand then unrolling, we produce the best-known colorings for several small values of rand k.

Despite the extreme success of spectral graph theory, there are relatively few papers applying spectral analysis to hypergraphs. Chung first introduced Laplacians for regular hypergraphs and showed some useful applications. Other researchers have treated hypergraphs as weighted graphs and then studied the Laplacians of the corresponding weighted graphs. In this paper, we aim to unify these very different versions of Laplacians for hypergraphs. We introduce a set of Laplacians for hypergraphs through studying high-order random walks on hypergraphs. We prove that the eigenvalues of these Laplacians can effectively control the mixing rate of high-order random walks, the generalized distances/diameters, and the edge expansions.

Are biological networks different from other large complex networks? Both large biological and nonbiological networks exhibit power-law graphs (number of nodes with degree k, N(k)~k-β), yet the exponents, β, fall into different ranges. This may be because duplication of the information in the genome is a dominant evolutionary force in shaping biological networks (like gene regulatory networks and protein-protein interaction networks) and is fundamentally different from the mechanisms thought to dominate the growth of most nonbiological networks (such as the Internet). The preferential choice models used for nonbiological networks like web graphs can only produce power-law graphs with exponents greater than 2. We use combinatorial probabilistic methods to examine the evolution of graphs by node duplication processes and derive exact analytical relationships between the exponent of the power law and the parameters of the model. Both full duplication of nodes (with all their connections) as well as partial duplication (with only some connections) are analyzed. We demonstrate that partial duplication can produce power-law graphs with exponents less than 2, consistent with current data on biological networks. The power-law exponent for large graphs depends only on the growth process, not on the starting graph.

We consider the diameter of a random graph G(n,p) for various ranges of p close to the phase transition point for connectivity. For a disconnected graph G, we use the convention that the diameter of G is the maximum diameter of its connected components. We show that almost surely the diameter of random graph G(n,p) is close to lognlog(np) if np→∞. Moreover if nplogn=c>8, then the diameter of G(n,p) is concentrated on two values. In general, if nplogn=c>c0, the diameter is concentrated on at most 2⌊1/c0⌋+4 values. We also proved that the diameter of G(n,p) is almost surely equal to the diameter of its giant component if np>3.6.

We propose a random graph model which is a special case of sparserandom graphs with given degree sequences which satisfy a power law. This model involves only a small number of paramo eters, called logsize and log-log growth rate. These parameters capture some universal characteristics of massive graphs. From these parameters, various properties of the graph can be derived. For example, for certai n ranges of the parameters, we wi II compute the expected distribution of the sizes of the connected components which almost surely occur with high probability. We illustrate the consistency of our model with the behavior of some massive graphs derived from data in telecommunications. We also discuss the threshold function, the giant component, and the evolution of random graphs in this model.

Let tr(n, r+1) denote the smallest integer msuch that every r-uniform hypergraph on nvertices with m+1 edges must contain a complete graph on r+1 vertices. In this paper, we prove thatlimn→∞t3(n, 4)(n3)⩽3+1712=0.593592….

One of the most vexing challenges of working with graphical structures is that most algorithms scale poorly as the graph becomes very large. The computation is extremely expensive even for polynomial algorithms, thus making it desirable to devise fast approximation algorithms. We herein propose a framework using advanced tools 1-6from random graph theory and spectral graph theory to address the quantitative analysis of the structure and dynamics of large-scale networks. This framework enables one to carry out analytic computations of observable network structures and capture the most relevant and refined quantities of realworld networks.