Dual phase evolution (DPE) is a process that drives self-organization within complex adaptive systems. It arises in response to phase changes within the network of connections formed by a system's components. DPE occurs in a wide range of physical, biological and social systems. Its applications to technology include methods for manufacturing novel materials and algorithms to solve complex problems in computation.

Dual phase evolution (DPE) is a process that promotes the emergence of large-scale order in complex systems. It occurs when a system repeatedly switches between various kinds of phases, and in each phase different processes act on the components or connections in the system. DPE arises because of a property of graphs and networks: the connectivity avalanche that occurs in graphs as the number of edges increases.

Social networks provide a familiar example. In a social network the nodes of the network are people and the network connections (edges) are relationships or interactions between people. For any individual, social activity alternates between a local phase, in which they interact only with people they already know, and a global phase in which they can interact with a wide pool of people not previously known to them. Historically, these phases have been forced on people by constraints of time and space. People spend most of their time in a local phase and interact only with those immediately around them (family, neighbors, colleagues). However, intermittent activities such as parties, holidays, and conferences involve a shift into a global phase where they can interact with different people they do not know. Different processes dominate each phase. Essentially, people make new social links when in the global phase, and refine or break them (by ceasing contact) while in the local phase.

The following features are necessary for DPE to occur.

DPE occurs where a system has an underlying network. That is, the system's components form a set of nodes and there are connections (edges) that join them. For example, a family tree is a network in which the nodes are people (with names) and the edges are relationships such as "mother of" or "married to". The nodes in the network can take physical form, such as atoms held together by atomic forces, or they may be dynamic states or conditions, such as positions on a chess board with moves by the players defining the edges.

In mathematical terms (graph theory), a graph ${\displaystyle \textstyle G=\langle N,E\rangle }$ is a set of nodes ${\displaystyle \textstyle N}$ and a set of edges ${\displaystyle \textstyle E\subset \{(x,y)\mid x,y\in N\}}$. Each edge ${\displaystyle \textstyle (x,y)}$ provides a link between a pair of nodes ${\displaystyle \textstyle x}$ and ${\displaystyle \textstyle y}$. A network is a graph in which values are assigned to the nodes and/or edges.

Graphs and networks have two phases: disconnected (fragmented) and connected. In the connected phase every node is connected by an edge to at least one other node and for any pair of nodes, there is at least one path (sequence of edges) joining them.

The Erdős–Rényi model shows that random graphs undergo a connectivity avalanche as the density of edges in a graph increases. This avalanche amounts to a sudden phase change in the size of the largest connected subgraph. In effect, a graph has two phases: connected (most nodes are linked by pathways of interaction) and fragmented (nodes are either isolated or form small subgraphs). These are often referred to as global and local phases, respectively.